A&A 572, A35 (2014) DOI: 10.1051/0004-6361/201423814 & c ESO 2014

Separating -giant and ice-giant by halting pebble

M. Lambrechts1, A. Johansen1, and A. Morbidelli2

1 Lund Observatory, Department of Astronomy and Theoretical , Lund University, Box 43, 22100 Lund, Sweden e-mail: [email protected] 2 Dep. Lagrange, UNSA, CNRS, OCA, Nice, France

Received 15 2014 / Accepted 25 August 2014


In the giant planets come in two flavours: gas giants ( and ) with massive gas envelopes, and ice giants ( and ) with much thinner envelopes around their cores. It is poorly understood how these two classes of planets formed. High accretion rates, necessary to form the cores of giant planets within the -time of protoplanetary discs, heat the envelope and prevent rapid gas contraction onto the core, unless accretion is halted. We find that, in fact, accretion of pebbles (∼cm sized particles) is self-limiting: when a core becomes massive enough it carves a gap in the pebble disc. This halt in subsequently triggers the rapid collapse of the super-critical gas envelope. Unlike gas giants, ice giants do not reach this threshold and can only bind low-mass envelopes that are highly enriched by vapour from sublimated icy pebbles. This offers an explanation for the compositional difference between gas giants and ice giants in the solar system. Furthermore, unlike -driven accretion scenarios, our model allows core formation and envelope attraction within disc life-times, provided that in protoplanetary discs are predominantly made up of pebbles. Our results imply that the outer regions of planetary systems, where the mass required to halt pebble accretion is large, are dominated by ice giants and that gas-giant in wide are enriched by more than 50 of solids. Key words. planets and : formation – planets and satellites: gaseous planets – planets and satellites: composition – planets and satellites: interiors – protoplanetary disks

1. Introduction by a factor of 10, Kobayashi et al. 2011) have been proposed in order to form the cores of the giant planets. For the gas giants, In the core accretion scenario (Pollack et al. 1996), giant plan- planetesimal accretion is then halted artificially, or the opacity ets form by attracting a gaseous envelope onto a core of in the envelope is lowered, in order to reduce the envelope at- and ice. This theory is supported by the large amount of heavy traction timescale (Hubickyj et al. 2005). The ice giants are en- elements – elements with atomic number above He – found in visioned to remain small, because the protoplanetary gas disc the giant planets in our solar system (Guillot 2005). Additional dissipates during slow envelope growth (Pollack et al. 1996; evidence is provided by the observed dependence of giant exo- Dodson-Robinson & Bodenheimer 2010). occurrence on the host , which is a proxy In this paper, we investigate the attraction of the gaseous for the dust mass enrichment of the protoplanetary disc (Fischer envelope when growth occurs by the accretion of pebbles, & Valenti 2005; Buchhave et al. 2012). as opposed to . Pebble accretion rates are suffi- However, from a theoretical perspective it is poorly under- ciently high to form the cores of giant planets in less than stood how the core accretion scenario could have taken place, if 1 million , even in wide orbits (Lambrechts & Johansen the cores grew by accretion of km-sized planetesimals and their 2012). Previous studies (Johansen & Lacerda 2010; Ormel & fragments. Protoplanetary disc life-times range from ∼3 Myr Klahr 2010; Bromley & Kenyon 2011; Lambrechts & Johansen (Haisch et al. 2001; Soderblom et al. 2013) to possibly as long 2012; Morbidelli & Nesvorny 2012) demonstrate that this is the as ∼6 Myr (Bell et al. 2013). This is much shorter than the time result of gas operating on pebbles, which dramatically in- needed to grow cores to completion in numerical simulations creases the accretion cross section (Sect. 2). The rapid accretion (Levison et al. 2010) of discs with solid surface com- of pebbles leads to high accretion luminosities that support a parable to the solar (MMSN, Hayashi growing gaseous envelope around the core (Sect. 3). We pro- 1981). Additionally, the gaseous envelope grows only slowly ceed by calculating the critical core mass, the lowest mass for on Myr timescales, because of the continued heating by accre- which a core can no longer sustain the hydrostatic balance of the tion of remnant planetesimals, even after clearing most of its proto-envelope. The critical core masses we find are of the order feeding zone (Pollack et al. 1996; Ikoma et al. 2000). Therefore, of ≈100 Earth masses (ME), too large compared to the inferred planets with gaseous envelopes are difficult to form by - core masses of the gas giants in the solar system. Fortunately, imal growth within ∼10 Myr, especially outside the current we find that there is a threshold mass already around 20 ME, of Jupiter (5 AU), where core growth timescales rapidly increase where the core perturbs the gas disc and halts the accretion of (Dodson-Robinson et al. 2009). pebbles, which initiates the collapse of the envelope before the As a result, protoplanetary discs with strongly enhanced critical core mass is reached (Sect. 4). This threshold mass is solid surface densities in planetesimals (exceeding the MMSN reached by the cores of the gas giants, but not by the ice giants in

Article published by EDP A35, page 1 of 12 A&A 572, A35 (2014) wider orbits. By combining our calculations of the pebble isola- 10−2 tion mass and the critical core mass as a function of the envelope 80% 60% 40% 20% 0% enrichment, we can make estimates of the bulk heavy element 10−3 content of the giant planets. We find a good agreement with the composition of the giant planets in the solar system (Sect. 5). −4 We also discuss the implications of our model on the occurrence 10 5AU ⋅ and composition of giant exoplanets (Sect. 6). Finally, we briefly Mpeb summarize our work (Sect. 7). 10−5 30AU 5AU /yr) E

2. Pebble accretion /(M −6 ⋅ 10 M ⋅ 30AU The pebble accretion scenario, as outlined in Lambrechts & Mplan Johansen(2012), starts with the growth of pebbles from the 10−7 initial grains embedded in the protoplanetary disc (with sizes

≈µm) by collisions (Birnstiel et al. 2012) or through sublima- −8 τ <τ 10 τacc τdisc tion and condensation cycles around ice lines (Ros & Johansen acc> disc 2013). A fraction of the population of pebbles that drift towards the host star form dense swarms that subsequently collapse un- 10−9 der self- to create planetesimals 100−1000 km in size. 10−1 100 101 102 103 M Such concentrations can occur through the , c/ME driven by the mutual drag between particles and gas (Youdin & Goodman 2005; Youdin & Johansen 2007; Johansen & Youdin Fig. 1. Pebble accretion rates (red), planetesimal accretion rates (grey), 2007), or for example through the presence of vortices (Barge and minimal accretion rates required to sustain a stable gas envelope & Sommeria 1995) or bumps (Whipple 1972). A more (black), as a function of the core mass. The curves for the minimal ac- cretion rates are nearly independent of orbital radius between 5−30 AU, detailed discussion can be found in the reviews by Chiang & but depend strongly on the opacity (Appendix B.3) and on the level of Youdin(2010) and Johansen et al.(2014). Finally, the largest envelope pollution by sublimation of icy pebbles. Labels at the top of planetesimals can act as the seeds of the planetary cores which the figure indicate H2O pollution of the as a percentage with grow by rapidly sweeping up the remaining pebbles (Lambrechts respect to pure H/He nebular gas, corresponding to the fraction 1 − β & Johansen 2012). from Eq. (11). The critical core mass to collapse the gas envelope can We consider here cores that grow predominantly by the ac- be found at the intersection of an accretion curve with a critical curve. cretion of particles with radii of approximately mm-cm. Particle Accretion rate curves that fall in the yellow dashed region are too slow sizes can be expressed as a function of the gas drag timescale (t ) to form the cores of the giant planets before the dissipation of the gas f ˙ and Keplerian frequency Ω in terms of the Stokes number disc (τacc = Mc/Mc > τdisc = 2 Myr). Red circles mark the mass above K which pebble accretion is halted (Eq. (12)) and the gravitational col- ρ R lapse of the gas envelope is triggered. τ = Ω t = • , (1) f K f ρH Here we have taken for simplicity an MMSN model with a par- where ρ• is the solid , R the particle radius, ρ the midplane √ gas density, and H the local gas scale height of the disc. Small ticle scale height given by Hp/H = αt/τ (Youdin & Lithwick dust particles (τ  1) are thus strongly coupled and comoving 2007), where αt is the turbulent diffusion parameter. Low parti- f cle scale heights are expected in dead-zones and discs where an- with the gas, while much larger objects (τf  1) are only weakly affected by gas drag. In the outer parts of the MMSN, in the gular momentum transport occurs primarily through disc winds region with semi-major axis a between 5 and 30 AU, particle (Turner et al. 2014). From particle stirring alone, scale heights of Hp/H ≈ 0.01 are expected (Bai & Stone 2010). In the MMSN, sizes between mm and cm correspond to τf ≈ 0.01−0.1. The seeds of the planetary cores accrete from the full scale the accretion rate when Hp < rH translates into height of pebbles at a rate !2/3 M  a −1 M M˙ = 80 c E , (4) ˙ c 6 Mc = 2rHΣpvH, (2) 1 M⊕ 10 AU 10 yr

(Ormel & Klahr 2010; Lambrechts & Johansen 2012). Here, Σp which is illustrated in Fig.1 (red curves). denotes for the surface density in pebbles and vH = rHΩK is the The growth of the core is driven by the radial drift of pebbles h i1/3 2 through the protoplanetary disc. Because of gas drag robbing Hill velocity at the Hill radius rH = GMc/(3ΩK) , with G the . Particles entering the have a pebbles of , they spiral towards the star with a velocity crossing time, τc ∼ rH/vH, comparable to the orbital timescale. Gas drag operates on pebbles on similar timescales, leading to v ≈ −2τ ηv , (5) their accretion by the core. This accretion rate does not depend r f K on the particle size between τf = 0.1−10, but moderately de- 2/3 for particles with τf < 1 (Weidenschilling 1977; Nakagawa et al. creases ∝(τf/0.1) for particles below τf = 0.1 (Lambrechts 1986). Here v is the Keplerian velocity and & Johansen 2012). We have assumed in Eq. (2) that the particle K scale height is smaller than the Hill radius of the core, which is 1  H 2 ∂ ln P  a 1/2 valid when core masses are larger than η = − ≈ 0.0015 (6) 2 a ∂ ln a AU  τ −3/2  α 3/2  a −3/4 M ≈ 0.19 f t M . (3) is the fraction by which the gas orbits slower than pure Keplerian c,2D 0.1 10−3 10 AU E , which is a function of the (local) gas scale height and

A35, page 2 of 12 M. Lambrechts et al.: Separating gas-giant and ice-giant planets by halting pebble accretion pressure gradient ∂ ln P/∂ ln a. Thus pebbles drift radially in- efficiency of fragments (Kobayashi et al. 2012). Global simula- wards within a short timescale, tions furthermore highlight that growth by fragments is ineffi- cient, because they get trapped in mean and  −1   3 τf r push planetary cores towards the star (Levison et al. 2010). To td ≈ 5.5 × 10 yr. (7) 0.1 10 AU overcome these issues, it has been proposed that fragmentation A core embedded in the disc can accrete a sizable fraction of this continues to mm-cm sizes (Ormel & Kobayashi 2012; Chambers radial pebble flux, f ≈ 30% (Morbidelli & Nesvorny 2012, see 2014). Pebbles, because of gas drag, do not suffer from destruc- also AppendixA). tive excitations or resonant trapping. In this work, we use surface densities in pebbles compara- ble to MMSN estimates, unless mentioned otherwise. This ap- proximation is supported by theoretical models of protoplane- 3. The growth of the proto-envelope tary discs that include dust growth by coagulation and radial drift 3.1. Accretion luminosity of particles (Brauer et al. 2008; Birnstiel et al. 2012). A more de- tailed discussion can be found in AppendixA. This approach can The attraction of the gaseous envelope is regulated by the growth also be justified observationally: measurements of the spectral of the solid core. Pebbles that down in the proto-atmosphere index of the dust opacity in protoplanetary discs reveal that a sig- of the core deposit their potential energy close to the core surface nificant fraction of solids grow to mm and cm sizes early on and which provides the heat necessary to support the envelope. The remain present over the life-time of the disc (Ricci et al. 2010). luminosity of the planet is thus a simple function of the accretion In an accompanying paper (Lambrechts & Johansen 2014) we rate, study pebble accretion on global scales, including dust coagula- M M˙ tion, pebble drift, and the growth of multiple cores. These results L = βG c c , (11) motivate the particle sizes and accretion rates used in this study. rc To conclude this section, we briefly highlight the main dif- ferences between planetesimal and pebble accretion, which al- where rc is the radius if the core (for an extended discussion ter the accretion luminosity of the core and therefore the criti- see AppendixB). Depending on the composition of the accreted cal core mass. The pebble accretion rate given in Eq. (2) is the material, a fraction 1 − β of the mass attracted by the planet is maximal possible one, because in this regime one accretes from lost by sublimation high up in the atmosphere and pollutes the the full Hill sphere, which is the largest possible gravitational envelope with material of high molecular (Hori & Ikoma reach of the core (Lambrechts & Johansen 2012). The accretion 2011). For this study we assume the bulk of the pebbles to be of of planetesimals, on the other hand, is significantly less efficient cometary composition, with a mass ratio of β = 0.5 of refractory compared to pebbles. The planetesimal accretion rate can be ex- elements to water ice. In Appendix B.3, we discuss in greater pressed as a fraction of the pebble accretion rate, detail the influence of the composition of the accreted material. Knowing the luminosity of the planet, we can now proceed to M˙ c,plan ≈ ψM˙ c,peb. (8) calculate the structure of the hydrostatic envelope surrounding the core. The efficiency of accretion, ψ, is equal to the ratio of the core radius rc to Hill radius rH, 3.2. The critical core mass !−1/3 r  a −1 ρ ψ c ≈ × −4 c , There exists a critical core mass where the inwards gravitational = 3 10 3 (9) rH 10 AU 5.5 g/cm pull of the core overcomes the pressure support by the released accretion heat and the envelope collapses. We numerically in- where ρ is the material density of the core. The reduced ac- c vestigate the envelope mass as a function of the accretion lumi- cretion rates are indicated by the grey lines in Fig.1. This re- nosity by constructing spherically symmetric envelopes in hy- sult follows from the assumption that the planetesimal velocity drostatic equilibrium (Figs. B.1 and B.2, detailed description in dispersion is equal to the Hill speed (Dodson-Robinson et al. AppendixB). We identify the critical core mass M as the 2009; Dodson-Robinson & Bodenheimer 2010), v = Ωr , and c,crit H H core mass for which we no longer find a hydrostatic solution gravitational focusing occurs from a radius (r /r )1/2r , which c H H (Mizuno 1980; Ikoma et al. 2000). This occurs in practice when is smaller than the planetesimal scale height H = v /Ω = r . plan H H the mass bound in the envelope is comparable to the mass of This leads to planetesimal accretion rates the core and the self-gravity of the gas atmosphere becomes important (Stevenson 1982). As a result the gas envelope falls M˙ c,plan ≈ rcΣpvH, (10) onto the core on the Kelvin-Helmholtz timescale (Ikoma et al. with Σp now the surface density in planetesimals. Collisional 2000; Piso & Youdin 2014, see also AppendixC). Earlier inves- fragments of planetesimals (0.1−1 km) have a reduced scale√ tigations show that higher accretion rates increase Mc,crit (Ikoma height and can be accreted more rapidly by a factor of 1/ ψ et al. 2000; Rafikov 2006). In this study we have broadened the (Rafikov 2004). range of accretion rates studied to include those from pebbles. Planetesimals are dynamically heated by the cores, which We also find that the critical core mass can be lowered signifi- triggers a fragmentation cascade. Because of efficient grinding cantly by increasing the mean molecular weight through subli- of planetesimals to dust, the planetesimal mass reservoir is re- mation of icy material in the deeper parts of the envelope where duced with time (Kobayashi et al. 2010; Kenyon & Bromley the temperature is above T ≈ 150 K, in agreement with pre- 2008). Therefore, core formation at 5 AU requires massive plan- vious studies (Stevenson 1982; Hori & Ikoma 2011). Figure1 etesimal discs, at least 10 times as massive as expected from the shows critical curves, in black, which connect the critical core MMSN (Kobayashi et al. 2011). At wider orbital radii no signif- mass to the accretion rate (for a standard opacity choice, see icant growth occurs, although pressure bumps caused by planet- also Appendix B.3). We find that, unless accretion is interrupted, triggered gap opening in the gas disc could increase the accretion pebble accretion leads to critical core masses &100 ME between

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Fig. 3. Deviation from the equilibrium sub-Keplerian velocity of the gas in the protoplanetary disc (vθ,gas/vK) due to the presence of a located at a normalized radius of r = 1 (here corresponding to an orbital radius of a = 5 AU). For cores with masses above 20 ME (blue curve) and higher (red and black curves) the gas orbits faster than the Keplerian velocity (horizontal dashed line) in a region outside the orbit of the planet. This ring acts as a trap for pebbles drifting inwards and halts the accretion of pebbles onto the core.

height of the disc and the planet’s location. A planet can perturb the gas-disc enough to make the gas velocity in a narrow ring just outside the orbit of the planet super-Keplerian (Paardekooper & Mellema 2006; Morbidelli & Nesvorny 2012). This happens Fig. 2. Gravitational of the midplane gas density from a when the planet is capable of perturbing significantly the density 50 ME-planet embedded in a protoplanetary disc. Displayed is a full an- of the disc, changing locally the sign of the radial density gra- nulus (θ = 2π-wide in azimuth) of the protoplanetary disc around the dient. In the ring where the gas is super-Keplerian, the action of planet located at normalized radius r = 1. The resulting pressure pertur- bation halts the radial migration of pebbles and thus the solid accretion gas-drag reverses. Pebbles are pushed outwards, instead of in- onto the core. The overdensities at radius r = 0.85 and r = 1.2 corre- wards. Thus, the pebbles have to accumulate at the outer edge spond to the regions with super-Keplerian rotation, that are highlighted of this ring, instead of migrating all the way through the orbit of in Fig.3. A Rossby , here centred at ( θ = 3.8, r = 1.2) has formed the planet. The accretion of pebbles by the planet now suddenly outside the orbit of the planet. stops. The enhancement in pebble density at the edge of the ring can possibly lead to the formation of new large planetesimals and even of new cores (Lyra et al. 2008; Kobayashi et al. 2012). 5−30 AU, which is an order of larger than the cores An isolation mass of about 50 M , for a disc with a in the solar system (Guillot 2005). E scale height of 5%, was previously suggested in Morbidelli & Nesvorny(2012). However, the authors used 2D simulations, 4. The pebble isolation mass which forced them to use a large smoothing parameter (equal to 60% of the planet’s Hill radius) in the planet’s gravitational We now highlight the existence of a limiting mass for giant plan- potential. The use of a large smoothing parameter weakens the ets above which no additional pebbles are accreted. Detailed gravitational perturbations of the planet on the disc, so that their 3D numerical simulations of an annulus of the protoplanetary estimate is probably an overestimate. disc show that as the planet grows larger than the pebble isola- To overcome this problem, we used here a new 3D ver- tion mass, sion of the code FARGO (Masset 2000; Lega et al. 2013). The  a 3/4 new code also handles the diffusion of energy and stellar irra- Miso ≈ 20 ME, (12) diation, but we used here its isothermal version for simplicity. 5 AU The 3D code adopts a cubic approximation for the gravitational local changes in the pressure gradient modify the rotation ve- potential of the planet (Kley et al. 2009), which is somewhat locity of the gas, which halts the drift of pebbles to the core equivalent to assuming a very small smoothing parameter in a (Eqs. (5), (6) and Figs.2,3). The value of the pebble isolation standard potential. mass depends dominantly on the orbital radius through the disc We modelled a disc from 0.625 to 1.62 in radius, the unit of 3 aspect ratio, Miso ∝ (H/a) , see also Sect. 4.1. Therefore pebble distance being the radius of the planet’s orbit, with an aspect ra- isolation becomes harder to attain at wider orbital separations in tio of 5% and a viscosity given by an α prescription (Shakura −3 flaring discs. & Sunyaev 1973) with αt = 6 × 10 . The radial bound- ary conditions were evanescent, which prevented reflection of 4.1. Calculation of the pebble isolation mass the spiral density wave. The boundary condition in co-latitude was instead reflecting. The resolution was 320 × 720 × 32 in We now determine at which mass a planet can isolate itself from the radial, azimuthal, and co-latitudinal directions, respectively. the flux of pebbles, and the dependence of the result on the scale We did a simulation with a planet of 20, 30, and 50 ME. The

A35, page 4 of 12 M. Lambrechts et al.: Separating gas-giant and ice-giant planets by halting pebble accretion simulations have been run for 60 orbits, when the disc seemed planetesimal accretion in setting the conditions for envelope col- to have reached a stationary structure. For the purpose of iden- lapse. Firstly, when a grows to a mass beyond the tifying the pebble isolation mass, it is not necessary to explic- pebble isolation mass, solid accretion will be abruptly termi- itly model the of particles between τf = 0.001−1 nated. The accretion luminosity is quenched and the critical core (Morbidelli & Nesvorny 2012). We therefore limit ourselves to mass drops to a value much smaller than the pebble isolation calculating the pressure perturbation able to halt the radial parti- mass (Fig.1), which triggers a of rapid gas accretion. cle drift (Eq. (5)). This is in sharp contrast with core growth by planetesimal ac- Figure3 shows the ratio between the azimuthal velocity of cretion, where the continuous delivery of solid material delays the disc and the Keplerian velocity as a function of radius. For the by millions of years (Pollack et al. each radius, the azimuthal velocity has been computed on the 1996). Halting planetesimal accretion to overcome this difficulty mid- of the disc; its average over the azimuth has been has been previously proposed (Hubickyj et al. 2005), but the mass-weighted. We also computed the vertically averaged az- formation of a clean gap in a planetesimal disc demands small imuthal velocity (all averages were mass-weighted) and found planetesimals with a low surface density (Shiraishi & Ida 2008), essentially the same radial profile. As one can see in the figure, which is inconsistent with models of core growth with planetes- away from the planet, the disc is uniformly sub-Keplerian (the imals (Levison et al. 2010; Kobayashi et al. 2011). azimuthal velocity is 0.9962 times the Kepler velocity). Instead, Secondly, the low value of the pebble isolation mass re- the planet induces strong perturbations in the gas azimuthal ve- solves an apparent paradox faced by any growth scenario for locity in its vicinity. In particular for the case of the 30 ME core, giant planets: the high accretion rates necessary to form cores there are two strong signatures, associated with the edges of the before gas dissipation results in critical core masses that are too shallow gap that the planet opens in the disc: a dip at r = 0.88 large by an order of magnitude (∼100 ME, Fig.1). Fortunately, where the gas is strongly sub-Keplerian, and a peak at r = 1.11 the self-shielding of pebble accretion yields much lower where the gas exceeds the Keplerian velocity (the azimuthal ve- core masses in 5−10 AU orbits. locity is 1.0025 times the Kepler velocity). In this situation, the Thirdly, the pebble isolation mass introduces a natural sharp pebbles are expected to stop drifting at r ≈ 1.15, where the divide of the giant planets into two classes: gas giants and ice gas turns from sub-Keplerian (beyond this distance) to super- giants. The latter category are those cores that did not reach the Keplerian (inside this distance). pebble isolation mass before disc dissipation. Therefore these Performing a simulation with a 50 ME planet we checked planets never stopped accreting pebbles during the life-time of that, as expected, the velocity perturbation is linear in the mass the gas disc and the resulting accretion heat prevents unpol- of the planet. Thus, the planet-mass threshold for turning the disc luted H/He envelopes from becoming unstable and undergoing barely super-Keplerian is Miso ≈ 20 ME. We verified this result runaway gas accretion. As a result, low-mass cores only con- with a simulation with a planet of this mass. tract low-mass envelopes with increased mean molecular weight We also checked, with a simulation with a 5 times smaller (Fig.1), which occurs after pebbles sublimate below the ice line value of the turbulent αt parameter, that the azimuthal velocity and the released water vapour is homogeneously distributed by has a negligible dependence on the viscosity of the disc. This . Formation of ice giants in this model is thus different was expected because for a disc undergoing perturbations by a from the classical core accretion scenario, where ice giants are small planet, the resulting disc structure is dominated by disc’s giant planets that had their growth prematurely terminated by internal pressure and hence its aspect ratio (Crida et al. 2006). In demanding gas dispersal during envelope contraction (Pollack completely inviscid discs, the estimate for the isolation mass is et al. 1996; Dodson-Robinson & Bodenheimer 2010). likely smaller by no more than a factor of ≈2 (Zhu et al. 2014). The dependence of Miso on the aspect ratio can be estimated 5. Pebble accretion in the solar system analytically. In fact, in the limit of negligible viscosity, scaling the disc’s aspect ratio H/a proportionally to the normalized Hill The pebble isolation mass increases with orbital radius (Eq. (12), radius of the planet rH/a, and adopting rH as basic unit of length, Fig.1), so the ice giants cannot have formed too close to the host the equations of motion for the fluid become independent of the star (<5 AU) where they would have become gas giant planets. planet mass (Korycansky & Papaloizou 1996). Given that the Such close formation distances for the ice giants are anyway not perturbation in azimuthal velocity is linear in the planet’s mass favoured in current models of the early migration of the (Korycansky & Papaloizou 1996), the result implies that the per- solar system (Walsh et al. 2011): following the gas giants, the ice turbation in azimuthal velocity has to be proportional to (H/a)3. giants migrated inwards, but remained outside 5 AU, and subse- With this result, we can now conclude with the dependency of quently moved outwards to distances between 11–17 AU which Miso on the location of the planet in a given disc. Assuming that are preferred to explain the late time orbital after disc a disc is flared like the MMSN, as H/a = 0.05(a/5 AU)1/4, we dissipation (Tsiganis et al. 2005). The formation of the ice giants obtain the result expressed in Eq. (12). Alternatively, for discs in our model is therefore compatible with the understanding of irradiated by the star the gas scale height goes as H/a ∝ a2/7 in the solar system. One intriguing option (Chiang & Goldreich 1997). The exact value of H/a depends on that was previously not possible, is the approximately in situ the level of viscous heating in the inner disc (Bitsch et al. 2013), formation of ice giants beyond 20 AU, since pebble accretion but for moderate disc accretion rates is similar to the MMSN is sufficiently fast. estimate at 5 AU. Therefore, the scaling for the isolation mass 6/7 in an irradiated disc takes the form Miso = 20 ME(a/5 AU) , which is slightly steeper than the MMSN-estimate. 5.1. Planetary composition By combining our calculations of the pebble isolation mass and 4.2. Implications of a pebble isolation mass the critical core mass (including the effect of pebble sublimation) we can calculate the heavy element mass fraction as a function The existence of this pebble isolation mass has three major of the total planet mass (full description in Appendix B.2). We implications that show the advantage of pebble accretion over do not compare our results directly with the inferred core masses

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60 10AU 60 30AU [%] [%] M M / / Z Z

20AU M 40 M 40


M M 20 iso 20 iso before isolation before isolation after isolation S after isolation S 0 J 0 J 10 100 1000 10 100 1000 M M /ME /ME Fig. 4. Total heavy element mass fraction as a function of the total Fig. 5. Similar to Fig.4, but representing planetary compositions of mass M of the giant planet, at different orbital radii (5, 10, 20, 30 AU). planets located at 5 and 30 AU (full and dashed lines respectively) for Planets that do not grow beyond the pebble isolation mass (red dots) different values of the accretion rate efficiency . The thick solid lines remain core-dominated, while those that grow larger will have most of correspond to the case where the full pebble accretion luminosity is re- their mass in gas. Estimates of the composition of Uranus and Neptune leased ( = 1) and the lower thin lines of the same colour to a 10 times (blue error bars, Helled et al. 2011) agree well with the prediction made smaller luminosity ( = 0.1). The red error bars, slightly displaced to in this paper for planets formed in the outer disc. Similarly, for the gas- the left for readability, give the heavy element mass, without the con- giant planets Jupiter and Saturn we find a good agreement between tribution of heavy elements added by late time gas accretion in a disc the 5−10 AU curves and the total heavy element mass estimated by with increased dust-to-gas ratio (Guillot & Hueso 2006), as suggested Guillot(2005), indicated by the orange error bars. In order to create this to explain the noble gas abundances in Jupiter. The orange downwards figure, we numerically calculated the composition of the planet when arrow is an upper limit on the heavy element content of Saturn if layered it becomes critical, taking into account the pollution of the envelope, convection occurs in the envelope (Leconte & Chabrier 2013). for planet masses below the pebble isolation mass (light blue curves). When the planet reached a mass larger than the pebble isolation mass, only nebular gas was added to continue growth (black curves). Here we present results with refractory fraction β = 0.5, but results depend weakly on the choice of β between 0.1−1. lower the accretion rate and thus the luminosity by the same fac- tor. These lower efficiencies do not change the fit for the solar system gas giants, while ice giants are in fact better matched, if they formed closer towards the star near the ice line, compared of the giant planets in the solar system, but instead with the to- to their current orbits. tal heavy element mass, as it is better constrained and transport could have occurred from core to envelope (Guillot 2005). We We briefly address two caveats concerning the measured find good agreement for Jupiter and Saturn between 5−10 AU, heavy element masses of the gas giants. Recently, it has while Uranus and Neptune could have formed at similar or wider been suggested that layered convection is important for Saturn orbits, as illustrated in Fig.4. (Leconte & Chabrier 2013), which allows for a higher total heavy element mass (an upper limit on the heavy element con- In contrast, the composition of the giant planets is difficult to tent is indicated by the downwards arrow). Also it is possible reproduce with planetesimal accretion. For ice giants, planetesi- that a fraction of the heavy elements gets delivered through gas mal accretion is too slow at wide orbits, which would make the accretion at a time close to the dissipation of the protoplanetary critical core mass too low (Fig.1). For gas giants, late-time plan- disc, which would explain the noble gas abundances of Jupiter. etesimal accretion after runaway gas accretion cannot add the In Fig.5, the red error bars are corrected for the heavy elements significant mass fraction of heavy elements in their envelopes, accreted in this stage after pebble accretion. because planetesimal capture rates are small (Guillot & Gladman 2000). Our encouraging correspondence between the model and the composition of the giant planets does not depend strongly on 6. Exoplanets the assumptions made on the composition or surface density of pebbles. For Fig.5 we have repeated our analysis, but with Our proposed formation model for giant planets is applicable to a 10 times lower accretion luminosity, by introducing a fudge extrasolar planetary systems as well. If a core reaches the pebble factor  = 0.1 into Eq. (11). Such a reduced luminosity could for isolation mass before disc dissipation, it becomes a . example be the result of the bulk composition of the pebbles to However, outside an orbital radius alim the growth of cores will be largely in ice, reducing the refractory fraction β by a factor be too slow and isolation masses too high for pebble isolation to of . Or, alternatively, the consequence of a lower surface den- occur within the disc life-time τdisc. Those cores end up as ice sity in pebbles compared to an MMSN-estimate, which would giants. In a disc with a solid surface density equal to the MMSN

A35, page 6 of 12 M. Lambrechts et al.: Separating gas-giant and ice-giant planets by halting pebble accretion

( = 1), we find the limiting semi-major axis for gas giants to be supported by accretion heat and in hydrostatic balance. This of- fers an explanation for the occurrence of planets that only at- !4/5 τ tract a thin envelope of and , mixed with large a ≈ 95 4/5 disc AU. (13) lim 2 Myr amounts of water vapour released by sublimation of icy pebbles, such as the ice giants Uranus and Neptune in our solar system. We demonstrate that this single model explains the bulk In massive protoplanetary discs,  is near unity or larger and alim is thus located far out in the disc. In such a case, wide-orbit gas composition of all the giant planets in the solar system. giants, such as the four planets around the A star HR8799 lo- Additionally, the pebble accretion scenario can be tested by cated at a = 15−70 AU (Marois et al. 2010), can form in situ. studying systems. We find that most exoplanets in wide orbits will be similar to ice giants. Only when cores grow We predict that their cores will be large (∼50−100 ME), because of the high pebble isolation mass at large orbital distances (or very large (&50 ME), within the disc life-time, can gas giants from the similarly large critical core masses for pure H/He en- form in wide orbits. We therefore predict a high solid enrich- velopes as seen in Fig.1). Such solid-enriched compositions are ment for gas-giant exoplanets in wide orbits, like those found in supported by models of gas-giant exoplanets now in close orbits the HR8799 . (but which likely formed at wider orbits) that show total heavy Acknowledgements. element masses ∼100 ME, much larger than planetesimal isola- M.L. thanks K. Ros, B. Bitsch and T. Guillot for comments. tion masses in classical core accretion (Pollack et al. 1996). For The authors also wish to thank the referee for thoughtful feedback which helped improve the manuscript. A.J. and M.L. are grateful for the financial support from example, the gas giant CoRoT-10 b has a mass of 870 ME, of the Royal Swedish Academy of Sciences and the Knut and Alice Wallenberg which approximately 180 ME are in heavy elements (Bonomo Foundation. A.J. was also acknowledges funding from the Swedish Research et al. 2010). Similarly, Corot-13 b, 14 b, 17 b, 20 b, and 23 b all Council (grant 2010-3710) and the European Research Council (ERC Starting Grant 278675-PEBBLE2PLANET). A.M. thanks the French ANR for support have &70 ME of heavy elements and are significantly enriched with respect to the host star metallicity (Moutou et al. 2013). on the MOJO project. Nevertheless, giant planets at wide orbits will more com- monly be ice giants. Indeed relaxing the efficiency somewhat Appendix A: Pebble surface density and accretion results in, for example, alim = 15 AU for  = 0.1 (in good agreement with the solar system). This seems also to be broadly efficiency consistent with the low occurrence of gas giants at very wide The pebble surface densities inferred from the minimum mass wide orbits inferred from direct imaging surveys (the fraction of solar nebula, used throughout this paper, are broadly consistent FGKM with planetary companions &2 Jupiter masses be- with surface densities calculated in more detailed models of pro- yond 25 AU is below 20%, Lafrenière et al. 2007). Altogether, toplanetary discs that combine dust growth and the drift of peb- pebble-driven envelope attraction predicts an orbital and com- bles. The reconstruction of the MMSN is based on the question- positional dichotomy, similar to the solar system, between gas able assumption that planets grow in situ out of all the material giants and ice giants in extrasolar systems. which is available locally. In contrast, observed protoplanetary discs are larger and show temporal evolution in both gas and dust components. This evolution is understood to be the con- 7. Summary sequence of a drift-limited dust growth by coagulation (Brauer The model we propose can be summarized as follows. First, et al. 2008). As demonstrated by Birnstiel et al.(2012), the dom- rocky/icy particles grow throughout the disc by sticking colli- inant particle size is well characterized by an equilibrium be- sions and condensation around ice lines. When solids reach mm- tween the local growth time scale and the drift timescale result- cm sizes they start to decouple from the gas in the protoplanetary ing in a narrow size range between τf = 0.01−0.1 at distances disc and drift towards the . Then, hydrodynamical concentra- between 5−30 AU. The surface density of these pebbles also re- tion mechanisms of pebbles, such as the streaming instability or mains high during the disc life-time. During the first ≈1 Myr the vortices, lead to the formation of a first generation of planetes- initial dust-to-gas ratio of 0.01 is maintained, only to decay to imals. Subsequently, the largest of these planetesimals continue ≈0.001 after ≈3 Myr. Therefore, the employed surface densities to grow by accreting from the flux of pebbles drifting through in the paper are adequate, certainly given the short timescales the disc. The high accretion rates result in core formation on a on which cores grow by pebble accretion. Furthermore, we also short timescale, within the disc life-time. Such fast growth im- demonstrate our model holds for surface densities reduced by an plies that the around the growing cores are strongly order of magnitude (Fig.5). supported by the accretion heat. The total pebble mass in the disc needed for our model is We show that the evolution of the gas envelope is different also consistent with protoplanetary disc observations. Pebbles depending on whether the core reaches the pebble isolation mass from the drift equilibrium model are efficiently accreted at a rate (or not), resulting in respectively gas giants or ice giants. When  2/3 a core grows sufficiently large, around 20 M at 5 AU, it can halt τf E M˙ = 2 r Σ v (A.1) accretion of solids onto the core by gravitationally perturbing c 0.1 H p H the surrounding gas disc. This creates a pressure bump that traps incoming pebbles. When the core gets isolated from pebbles, as demonstrated by numerical simulations for particles in the the envelope is no longer hydrostatically supported by accretion τf = 0.01−0.1 range (Lambrechts & Johansen 2012). The radial heat, and gas can be accreted in a runaway fashion. This leads to flux of pebbles through the disc is given by the formation of gas giants like Jupiter and Saturn. ˙ Cores in wider orbits need to grow more massive than 20 ME Mdrift = 2πaΣpvr ≈ 4πΣdaτfηvK, (A.2) to reach isolation, because of the steep increase in the gas scale height in flaring discs. Therefore, wide-orbit cores that do not where vK is the Keplerian velocity at orbital radius a and η is a grow larger than 50−100 ME during the gas disc phase remain dimensionless measure of radial gas pressure support (Eq. (6)).

A35, page 7 of 12 A&A 572, A35 (2014)

The embryo will accrete the fraction f = M˙ c/M˙ drift of these interior to a radius r is given by M(r). Mass continuity is guar- solids, anteed by 20  τ −1/3 r 2/3 dM(r) f ≈ η−1 f H (A.3) = 4πr2ρ. (B.2) 4π 0.1 a dr !2/3  τ −1/3 M  a −1/2 = 0.35 f c · (A.4) Energy can be transported either by radiation diffusion or con- 0.1 20 ME 5 AU vection in optically thick regions. Convective heat transport is triggered when The filtering factor f itself does not explicitly depend on Σd, η ∂ ln T γ − 1 but does depend on , thus in regions with reduced pressure > , (B.3) support (pressure bumps) the efficiency could be higher. The ∂ ln P γ necessary mass in pebbles in the disc can be estimated from −1 where T is the local temperature and γ is the adiabatic index. dMp = f dMc, Convective transport takes the form !1/3  1/2  1/3 Mc a τf dT γ − 1 T GM(r)ρ Mp ≈ 165 ME , (A.5) = − , (B.4) 20 ME 5 AU 0.1 dr γ P r2 in order to grow the core to 20 ME (starting from 0.1 ME, but while radiative transport depends on the opacity κ and the lumi- the integral only weakly depends on this choice). Therefore, the nosity L, mass reservoir in the outer disc is on this order and the total disc mass should be about 0.05 solar mass (for a standard Z = 0.01 dT 3 κLρ = − · (B.5) metallicity). Larger disc masses are not needed to form more dr 64πσ r2T 3 planets in the disc, because of the low value of f for small core masses. The Stefan-Boltzmann constant is denoted by σ. The equation The disc masses needed for our model are consistent with of state (EoS) observations. The best studied protoplanetary disc, the disc of kB the star TW Hydrae, has a gas mass of 0.05 M (Bergin et al. P = ρT, (B.6) µ 2013) and the mm and gas distribution are well described by the model of Birnstiel et al.(2012). From mm-surveys it seems with kB the Boltzmann constant and µ the mean molecular such disc masses for solar like stars show a large spread between weight, relates the pressure to the density and closes the system −4 −1 10 –10 M (Andrews et al. 2013) and therefore the disc mas of equations. in our model lies in the higher end of this distribution. However, In principle, one could solve for an energy equation, disc mass estimates are based on an assumed ratio of gas to mm- sized dust of 100, and therefore these mass estimates may be dL = 4πr2ρ (B.7) lower limits. dr The above analysis does not take into account the presence of ice lines. In these regions particle sizes and the local surface where  is the heat deposited at radius r. However, potential en- density are set by a condensation-sublimation cycle across the ergy of accreted material is deposited deep in the convective in- ice line, resembling hail formation (Ros & Johansen 2013). This terior close to the core surface, so we take a constant luminosity is very different from the coagulation-drift equilibrium situation as a function of planetary radius L(r) = L. When pebbles settle discussed above. Likely, around the various ice lines in proto- with terminal velocity in the atmosphere, drag counterbalances gravity and locally deposits frictional heat per unit mass plantary discs (H2O, CO2, CO) solid surface densities in pebbles of rather large size (τf ∼ 0.1) greatly exceed MMSN estimates, GM δE ≈ c δr. promoting fast core growth. Furthermore, recondensation of sub- 2 (B.8) limated pebbles onto particles exterior of the ice line reduces the r loss of ices. Therefore, per unit length, the deposited energy is much larger A more thorough discussion of core growth can be found in close to the core surface than in the upper atmosphere, by a factor Lambrechts & Johansen(2014), where we investigate embryo of 106 for the atmospheres studied here. The luminosity profile growth in a global model that includes dust growth and the drift takes the form of pebbles. GM M˙ GM M˙  r  L(r) ≈ c − c = 1 − c L, (B.9) rc r r Appendix B: Calculating the critical core mass revealing that the luminosity deviates from the constant value B.1. Structure of the proto-envelope adopted here only near the core surface (Rafikov 2006). We numerically solve the standard equations for planetary atmo- Additionally, we ignore the heat from the contraction of the en- velope, the latent heat from evaporation, and nuclear heating spheres (Kippenhahn & Weigert 1990). The envelope is assumed 26 to be spherically symmetric and in hydrostatic balance, by the core through the decay of Al, for the following rea- sons. The luminosity generated from binding the gas envelope dP GM(r)ρ to the growing core can be ignored when the core is subcritical = − · (B.1) dr r2 (Rafikov 2006). We find that latent heat of water sublimation can only be important for small cores. For a certain accretion rate, we Here G is the gravitational constant, P the pressure, and ρ the can assume that of the accreted material a fraction β of refrac- density at position r from the centre of the planet. The mass tory grains settles to the core, while a remaining fraction 1 − β is

A35, page 8 of 12 M. Lambrechts et al.: Separating gas-giant and ice-giant planets by halting pebble accretion

a= 5 AU a=10 AU 105 M M c= 8ME c= 8ME

104 /K T 103


10−2 M M c= 8ME c= 8ME 10−4

−6 )

3 10

−8 /(g/cm 10 ρ



109 1010 1011 1012 1013 109 1010 1011 1012 1013 r/cm r/cm

Fig. B.1. Dependence of the hydrostatic envelope on orbital radius and envelope pollution. The left panels give the temperature (top) and density profile (bottom), for both a 50% polluted and unpolluted atmosphere (grey curve) of an 8 ME core accreting pebbles at 5 AU. Thick lines represent the regions where heat transport by convection dominates. The profile is given starting from the connection point to the Hill sphere. The yellow circle gives the of the envelope, while the black circle indicates the Bondi radius (the distance where the escape speed equals the local sound speed). The depth at which the envelope density is enhanced by water vapour is marked by the dashed blue lines in the lower panels. The 3 dotted line represents the core radius (assuming ρc = 5.5 g/cm ). The right column is similar, but for a planet orbiting at 10 AU. water ice that sublimates. The latent heat per unit mass required to take the self-gravity of the envelope into account until we con- for the sublimation of the water ice fraction is given by verge to a self-consistent solution. Additionally, we take into ac- − count the sublimation of ice from settling pebbles, by altering Q = −2.3 × 103(1 − β) J g 1. (B.10) sub the mean molecular weight and the equation of state below the The fraction that settles to the core gives ice line, under the assumption that convection causes an approx- imately homogeneous mixture. There the molecular weight, GMc 4 M −1 Qgrav = β = 6.3 × 10 β J g . (B.11) Rc ME 1 − X µ−1 = + , (B.12) So the latent heat becomes important for β = 0.1 and small mix µH/He µH2O cores (Mc . 1 ME). Finally, of short-lived radio isotopes in chondritic material from the core releases a lu- depends on the mass fraction of water vapour with respect to 24 minosity L = 1.5 × 10 (Mrock/ME) exp (−t/τ26Al) /s. Here, the solar nebula H/He mixture X, with µH/He = 2.34 mH and 26 τ26Al = 1.01 Myr is the decay time of Al. The importance of µH2O = 18 mH (mH is the mass of the H atom). In the con- this heat source depends on the time of giant planet formation vective interior, the calculation of the temperature gradient re- after the formation of CAI, but remains about 4 orders of mag- lies on the specific heat capacity of the mixture cP,mix = (1 − nitude smaller that the heat released by the accretion of pebbles. X)cP,H/He + XcP,H2O, with cP,H/He = [γH/He/(γH/He − 1)]kB/µH/He

In practice we integrate stepwise from the Hill sphere, where and cP,H2O = [γH2O/(γH2O − 1)]kB/µH2O. We find that the critical we assume nebular conditions (T0, ρ0), to the core surface in or- core mass is very sensitive to the mean molecular weight, but der to calculate the envelope structure. We iterate this procedure less so to the adiabatic index, which we we have taken here to

A35, page 9 of 12 A&A 572, A35 (2014)

a=20 AU a=30 AU 105 M M c= 8ME c= 8ME

104 /K T 103


10−2 M M c= 8ME c= 8ME 10−4

−6 )

3 10

−8 /(g/cm 10 ρ



109 1010 1011 1012 1013 109 1010 1011 1012 1013 r/cm r/cm

Fig. B.2. Comparison of the structure between the hydrostatic envelope enriched by H2O steam and a pure H/He atmospheres, around embryo cores of 8 ME located at 20 and 30 AU. Labels are similar to Fig. B.1.

be γH/He = 1.4 (appropriate for a diatomic gas) and γH2O = 1.17 core mass, as is standardly done (Mizuno 1980). The precise (for water steam at high T with all 12 degrees of freedom re- value of the critical core mass depends on the assumed accretion leased). More detailed calculations of the adiabatic index would rate, opacity, and composition of the envelope. require solving for multiple chemical species in the envelope. We have performed two classes of calculations of the critical Such calculations show that, in the limit of very polluted en- core mass. In the first class, we have kept the accretion rate con- velopes (∼90%) from accreted material with a -like com- stant during the iteration over the core masses, in order to decou- position, changes in γ can lead to reduced critical core masses ple the stability of an envelope from the assumed accretion rates by at most a factor of 2 (Hori & Ikoma 2011), which is an effect by either planetesimals or pebbles onto cores. The results are the also seen in our simplified model. black curves in Fig.1, where we explicitly showed the depen- At wide orbital distances where the density is low, the pho- dency on the assumed composition of the envelope. In the sec- tosphere is located below the Hill sphere and the region is ond class, we have self-consistently taken into account the de- T 4 T 4 L/ πσr2 nearly isothermal with = 0 + (16 )(Rafikov 2006). pendency of the mass accretion rate of pebbles, and thus the Examples of the envelope structure of planets located at various luminosity, on the mass of the planet. This is important in ob- orbital distances can be inspected in Figs. B.1 and B.2. We dis- taining the results displayed in Figs.4 and5, where we calculate cuss in greater detail the prescription of the opacity and the role the planetary composition as a function of mass. The curves in of dust grains in Sect. B.3. these figures are obtained by considering two regimes. In the first regime, the core has not reached isolation. Consequently, for a B.2. Determining the critical core mass given mass, and thus accretion rate, we find by iteration the level of enrichment in heavy elements in the atmosphere required to We find the critical core mass numerically by stepwise increas- collapse the envelope. The composition of the planet at this crit- ing the core mass. When we not longer find a hydrostatic solu- ical point is the one displayed by the ratio of the heavy elements, tion for the envelope, we identify this core mass as the critical in the core plus envelope, to the total planet mass.

A35, page 10 of 12 M. Lambrechts et al.: Separating gas-giant and ice-giant planets by halting pebble accretion

10−2 Appendix C: Envelope contraction −3 10 Finally, we briefly discuss the Kelvin-Helmholtz timescale on 10−4 which envelopes contract −5 /yr)

E 10 GMc Menv −6 τKH ≈ (C.1)

/(M 10 Reff L ⋅

M −7 10 with Reff some effective radius, typically taken to be the radius −8 10 5.0AU at the convective to radiative border (Ikoma et al. 2000; Pollack 10−9 et al. 1996). After pebble isolation, the high luminosity from 10−1 100 101 102 103 pebble accretion will almost instantaneously contract the core to M /M a state where only luminosity caused from envelope contraction c E is important. Following numerical results by Ikoma et al.(2000) Fig. B.3. Dependency of the critical core mass on the dust opacity in contraction occurs on a timescale H/He envelopes. Grey curves correspond to factor of ten different dust !−2.5 !1 M κ opacities to the opacity used in this study (solid black line). The critical τ × 5 c , KH = 3 10 2 −1 yr (C.2) curves are here given at 5 AU, but have little dependency on orbital 10 ME 1 cm g radius. for envelope masses comparable to core masses Mc. The depen- dency on the dust opacity κ means this is an upper limit in the case of pebble accretion, as after isolation few grains will be deposited in the envelope. Contraction could be delayed more In the second regime, the planet has grown beyond the iso- by continued solid accretion from planetesimals (Pollack et al. lation mass. In this case, the mass of heavy elements is taken to 1996). However, this can be ignored in the scenario described be equal to that found in the previous regime for a planet mass in the main paper, as firstly we do not assume all solid density equal to the pebble isolation mass. The remainder of the mass in planetesimals, and additionally we do not propose a signifi- of the planet is H/He from the gas accretion phase. Formally, at cant enrichment of the planetesimal column density (larger than isolation the envelope does not have to be polluted to start gas a factor of 5) as in Pollack et al.(1996). We emphasize that at accretion, so one could also choose to include only the heavy the brink of collapse the Hill sphere of a gas giant is comparable elements from the core, but this would constitute only a minor to the gas disc scale height, which makes the spherical approxi- correction to the composition compared to the total mass of the mation invalid. Future studies will have to explore the effects of giant planets. moving away from the standardly assumed spherical symmetry.

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